lilia rosati - Academia.edu (original) (raw)

Papers by lilia rosati

Topology and its Applications, 2012

Let M be a smooth manifold and let F be a codimension one, C ∞ foliation on M , with isolated sin... more Let M be a smooth manifold and let F be a codimension one, C ∞ foliation on M , with isolated singularities of Morse type. The study and classification of pairs (M, F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb [Reeb] states that a manifold admitting a foliation with exactly two centertype singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper [Ee-Kui] classify manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices). In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F , we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S 1 , we are able to extend Haefliger's theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result. Acknoledgements I am very grateful to prof. Bruno Scárdua for proposing me such an interesting subject and for his valuable advice. My hearthy good thanks to prof. Graziano Gentili for his suggestions on the writing of this article. 1 Foliations and Morse Foliations Definition 1.1 A codimension k, foliated manifold (M, F) is a manifold M with a differentiable structure, given by an atlas {(U i , φ i)} i∈I , satisfying the following properties: (1) φ i (U i) = B n−k × B k ; (2) in U i ∩ U j = ∅, we have φ j • φ −1 i (x, y) = (f ij (x, y), g ij (y)), where {f ij } and {g ij } are families of, respectively, submersions and diffeomorphisms, defined on natural domains. Given a local chart (foliated chart) (U, φ), ∀x ∈ B n−k and y ∈ B k , the set φ −1 (•, y) is a plaque and the set φ −1 (x, •) is a transverse section. The existence of a foliated manifold (M, F) determines a partition of M into subsets, the leaves, defined by means of an equivalence relation, each endowed of an intrinsic manifold structure. Let x ∈ M ; we denote by F x or L x the leaf of F through x. With the intrinsic manifold structure, F x turns to be an immersed (but not embedded, in general) submanifold of M. In an equivalent way, a foliated manifold (M, F) is a manifold M with a collection of couples {(U i , g i)} i∈I ,

Boundary Value Problems, 2013

In this note we consider the classical Massera theorem, which proves the uniqueness of a periodic... more In this note we consider the classical Massera theorem, which proves the uniqueness of a periodic solution for the Liénard equation x + f (x)ẋ + x = 0, and investigate the problem of the existence of such a periodic solution when f is monotone increasing for x > 0 and monotone decreasing for x < 0 but with a single zero, because in this case the existence is not granted. Sufficient conditions for the existence of a periodic solution and also a necessary condition, which proves that with this assumptions actually it is possible to have no periodic solutions, are presented. MSC: 34C05; 34C25

Nonlinear Analysis-theory Methods & Applications - NONLINEAR ANAL-THEOR METH APP, 2007

The phase portrait of the second-order differential equation ẍ+∑l=0nfl(x)ẋl=0, is studied. Some... more The phase portrait of the second-order differential equation ẍ+∑l=0nfl(x)ẋl=0, is studied. Some results concerning existence, non-existence and uniqueness of limit cycles are presented. In particular, a generalization of the classical Massera uniqueness result is proved.

Topology and its Applications, 2012

Let M be a smooth manifold and let F be a codimension one, C ∞ foliation on M , with isolated sin... more Let M be a smooth manifold and let F be a codimension one, C ∞ foliation on M , with isolated singularities of Morse type. The study and classification of pairs (M, F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb [Reeb] states that a manifold admitting a foliation with exactly two centertype singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper [Ee-Kui] classify manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices). In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F , we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S 1 , we are able to extend Haefliger's theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result. Acknoledgements I am very grateful to prof. Bruno Scárdua for proposing me such an interesting subject and for his valuable advice. My hearthy good thanks to prof. Graziano Gentili for his suggestions on the writing of this article. 1 Foliations and Morse Foliations Definition 1.1 A codimension k, foliated manifold (M, F) is a manifold M with a differentiable structure, given by an atlas {(U i , φ i)} i∈I , satisfying the following properties: (1) φ i (U i) = B n−k × B k ; (2) in U i ∩ U j = ∅, we have φ j • φ −1 i (x, y) = (f ij (x, y), g ij (y)), where {f ij } and {g ij } are families of, respectively, submersions and diffeomorphisms, defined on natural domains. Given a local chart (foliated chart) (U, φ), ∀x ∈ B n−k and y ∈ B k , the set φ −1 (•, y) is a plaque and the set φ −1 (x, •) is a transverse section. The existence of a foliated manifold (M, F) determines a partition of M into subsets, the leaves, defined by means of an equivalence relation, each endowed of an intrinsic manifold structure. Let x ∈ M ; we denote by F x or L x the leaf of F through x. With the intrinsic manifold structure, F x turns to be an immersed (but not embedded, in general) submanifold of M. In an equivalent way, a foliated manifold (M, F) is a manifold M with a collection of couples {(U i , g i)} i∈I ,

Boundary Value Problems, 2013

In this note we consider the classical Massera theorem, which proves the uniqueness of a periodic... more In this note we consider the classical Massera theorem, which proves the uniqueness of a periodic solution for the Liénard equation x + f (x)ẋ + x = 0, and investigate the problem of the existence of such a periodic solution when f is monotone increasing for x > 0 and monotone decreasing for x < 0 but with a single zero, because in this case the existence is not granted. Sufficient conditions for the existence of a periodic solution and also a necessary condition, which proves that with this assumptions actually it is possible to have no periodic solutions, are presented. MSC: 34C05; 34C25

Nonlinear Analysis-theory Methods & Applications - NONLINEAR ANAL-THEOR METH APP, 2007

The phase portrait of the second-order differential equation ẍ+∑l=0nfl(x)ẋl=0, is studied. Some... more The phase portrait of the second-order differential equation ẍ+∑l=0nfl(x)ẋl=0, is studied. Some results concerning existence, non-existence and uniqueness of limit cycles are presented. In particular, a generalization of the classical Massera uniqueness result is proved.